Product Approximations for Solutions to a Class of Evolution Equations in Hilbert Space
Résumé
In this article we prove approximation formulae for a class of unitary evolution operators $U (t,s)_{s,t \in \left[0,T\right]}$ associated with linear non-autonomous evolution equations of Schrödinger type defi
ned in a Hilbert space $\mathcal{H}$. An important feature of the equations we consider is that both the corresponding self-adjoint generators and their domains may depend explicitly on time, whereas the associated quadratic form domains may not. Furthermore the evolution operators we are interested in satisfy the equations in a weak sense. Under such conditions the approximation formulae we prove for $U(t,s)$ involve weak operator limits of products of suitable approximating functions taking values in $\mathcal{L(H)}$, the algebra of all linear bounded operators on $\mathcal{H}$. Our results may be relevant to the numerical analysis of $U(t,s)$ and we illustrate them by considering two evolution problems in quantum mechanics.
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